Static Zero Field Relaxation - Revision history

Jess at 18:25, 17 September 2022

2022-09-17T18:25:56Z

2022-09-15T22:06:43Z

2022-09-15T21:38:12Z

2022-09-15T18:00:51Z

2022-09-15T17:02:20Z

2022-09-15T17:01:39Z

2022-09-15T17:00:56Z

2022-09-14T23:24:50Z

2022-09-14T23:17:34Z

2022-09-14T23:14:12Z

@@ Line 16: / Line 16: @@
 <center>
-[[Image:KT-GZF.png|600px|inline image (click to see full size)]]
+[[Image:KT-GZF.png|400px|inline image (click to see full size)]]
 {| width="80%"
 ! align="center" |
@@ Line 30: / Line 30: @@
 <center>
-[[Image:Toshi-ZF.png|600px|inline image (click to see full size)]]
+[[Image:Toshi-ZF.png|400px|inline image (click to see full size)]]
 {| width="80%"
 ! align="center" | Yamazaki's generalized relaxation function <math>g_{zz}^{\rm TY}(n,\Delta,t)</math>

@@ Line 28: / Line 28: @@
 </center>
 This function should be regarded as strictly phenomenological, like the "''stretched exponential''" (which see).
+<center>
+[[Image:Toshi-ZF.png|600px|inline image (click to see full size)]]
+{| width="80%"
+! align="center" | Yamazaki's generalized relaxation function <math>g_{zz}^{\rm TY}(n,\Delta,t)</math>
+for power laws <math>n=1</math> (red), <math>n=2</math> (black) and <math>n=3</math> (blue).
+|}
+</center>

@@ Line 19: / Line 19: @@
 {| width="80%"
 ! align="center" |
-<math>g_{zz}^G(\Delta,t) = {1\over3} + {2\over3} \exp\left[{1\over2}\Delta^2 t^2\right]\left(1 - \Delta^2 t^2\right)</math>.
+<math>g_{zz}^G(\Delta,t) = {1\over3} + {2\over3} \left(1 - \Delta^2 t^2\right) \; \exp\left[\Delta^2 t^2 \over 2 \right]</math>.
 |}
 </center>
-The static ZF Kubo-Toyabe function was later generalized by Toshimitsu Yamazaki for field discributions representable by a generalized power law, namely
+The static ZF Kubo-Toyabe function was later generalized by Toshimitsu Yamazaki for field distributions representable by a generalized power law, giving
 <center>
-<math>{\cal D}_n^{\rm TY}(|B|) = {1 \over \sqrt{n\pi} ? \; \Delta} \exp\left[- {1\over n} \left(B \over \Delta \right)^n\right] </math>, ?? CHECK THIS!
+<math>g_{zz}^{\rm TY}(n,\Delta,t) = {1\over3} + {2\over3} \left(1 - \Delta^n t^n\right) \; \exp\left[\Delta^n t^n \over n\right]</math>.
-</center>
-giving
-<center>
-<math>g_{zz}^{\rm TY}(n,\Delta,t) = {1\over3} + {2\over3} \exp\left[{1\over n?}(\Delta t)^n\right]\left[1 - (\Delta t)^2\right]</math>.  ?? CHECK THIS!
 </center>
-This function should be regarded as strictly phenomenological, like the "''stretched exponential''", which see.
+This function should be regarded as strictly phenomenological, like the "''stretched exponential''" (which see).

@@ Line 22: / Line 22: @@
 |}
 </center>
+The static ZF Kubo-Toyabe function was later generalized by Toshimitsu Yamazaki for field discributions representable by a generalized power law, namely
+<center>
+<math>{\cal D}_n^{\rm TY}(|B|) = {1 \over \sqrt{n\pi} ? \; \Delta} \exp\left[- {1\over n} \left(B \over \Delta \right)^n\right] </math>, ?? CHECK THIS!
+</center>
+giving
+<center>
+<math>g_{zz}^{\rm TY}(n,\Delta,t) = {1\over3} + {2\over3} \exp\left[{1\over n?}(\Delta t)^n\right]\left[1 - (\Delta t)^2\right]</math>.  ?? CHECK THIS!
+</center>
+This function should be regarded as strictly phenomenological, like the "''stretched exponential''", which see.

@@ Line 18: / Line 18: @@
 [[Image:KT-GZF.png|600px|inline image (click to see full size)]]
 {| width="80%"
-! align="left" |
+! align="center" |
 <math>g_{zz}^G(\Delta,t) = {1\over3} + {2\over3} \exp\left[{1\over2}\Delta^2 t^2\right]\left(1 - \Delta^2 t^2\right)</math>.
 |}

@@ Line 13: / Line 13: @@
 <math>{\cal D}(|B|) = {1 \over \sqrt{2\pi} \; \Delta} \exp\left[- {1\over2} \left(B \over \Delta \right)^2\right] </math>,
 </center>
-the relaxation function (named after Kubo and Toyabe, who first described it) looks like this:
+the resultant "Kubo-Toyabe" static ZF relaxation function (named after Kubo and Toyabe, who first described it) looks like this:
 [[Image:KT-GZF.png|600px|inline image (click to see full size)]]